Ban--Linial's Conjecture and treelike snarks

TL;DR

This paper proves that treelike snarks, a special class of complex cubic graphs with high circular flow number and excessive index, admit a 2-bisection, advancing understanding of Ban--Linial's Conjecture.

Contribution

It demonstrates that treelike snarks, which are critical in graph theory conjectures, admit a 2-bisection, providing new evidence related to Ban--Linial's Conjecture.

Findings

Treelike snarks admit a 2-bisection.

Treelike snarks have circular flow number at least 5.

Treelike snarks have excessive index at least 5.

Abstract

A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well-known Petersen graph, admits a 2-bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection. The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are known as snarks, in particular, those with excessive index at least 5, that is, whose edge set cannot be covered by four perfect matchings. Moreover, in [J. Graph Theory, 86(2) (2017), 149--158], Esperet et al. state that a possible counterexample to Ban--Linial's Conjecture must have circular flow number at least 5. The same authors also state that although empirical evidence shows that several graphs obtained from the Petersen graph admit a 2-bisection, they can offer nothing in the direction of a general proof. Despite some sporadic computational results, until now, no general result about snarks having excessive index and circular flow number both at least 5 has been proven. In this work we show that treelike snarks, which are an infinite family of snarks heavily depending on the Petersen graph and with both their circular flow number and excessive index at least 5, admit a 2-bisection.